Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?

For Balmer series: $\frac{1}{\lambda }=R\left(\frac{1}{2^2}-\frac{1}{n^2}\right),n=3,4,5,...$

$\frac{1}{{\lambda }_{max}}=R\left(\frac{1}{4}-\frac{1}{9}\right)=\frac{5R}{36}$

$\frac{1}{{\lambda }_{min}}=\frac{R}{4}$

$\Rightarrow \frac{{\lambda }_{\max }}{{\lambda }_{\min }}=\frac{\left(\frac{R}{4}\right)}{\left(\frac{5R}{36}\right)}=\frac{9}{5}$                     Option (a) is correct.

For Lyman series:$\frac{1}{\lambda }=R\left(1-\frac{1}{n^2}\right),n=2,3,\mathrm{4....}$

$\frac{1}{{\lambda }_{\min }}=R$

$\Rightarrow {\lambda }_{\min }={\lambda }_0=\frac{1}{R}$

$\frac{1}{\lambda }=\frac{1}{{\lambda }_0}\left(1-\frac{1}{n^2}\right)$

$\Rightarrow \lambda =\frac{{\lambda }_0{n}^2}{n^2-1}$                         Option (c) is wrong.

For Lyman series:

${\left(\frac{1}{{\lambda }_{\max }}\right)}_{\text{Lyman}}=R\left(\frac{1}{1^2}-\frac{1}{2^2}\right)\Rightarrow {\left({\lambda }_{\max }\right)}_{\text{Lyman}}=\frac{4}{3R}=\frac{1.33}{R}$

${\left(\frac{1}{{\lambda }_{\min }}\right)}_{\text{Lyman}}=R\left(\frac{1}{1^2}-\frac{1}{{\infty }^2}\right)\Rightarrow {\left({\lambda }_{\min }\right)}_{\text{Lyman}}=\frac{1}{R}$

So, Lyman series wavelengths vary between $\frac{1}{R}$ to $\frac{1.33}{R}$

For Balmer series:

${\left(\frac{1}{{\lambda }_{\max }}\right)}_{\text{Balmer}}=R\left(\frac{1}{2^2}-\frac{1}{3^2}\right)\Rightarrow {\left({\lambda }_{\max }\right)}_{\text{Balmer}}=\frac{36}{5R}=\frac{7.2}{R}$

${\left(\frac{1}{{\lambda }_{\min }}\right)}_{\text{Balmer}}=R\left(\frac{1}{2^2}-\frac{1}{{\infty }^2}\right)\Rightarrow {\left({\lambda }_{\min }\right)}_{\text{Balmer}}=\frac{4}{R}$

So, Balmer series wavelengths vary between $\frac{4}{R}$ to $\frac{7.2}{R}$

For Paschen series:

${\left(\frac{1}{{\lambda }_{\max }}\right)}_{\text{Paschen}}=R\left(\frac{1}{3^2}-\frac{1}{4^2}\right)\Rightarrow {\left({\lambda }_{\max }\right)}_{\text{Paschen}}=\frac{144}{7R}=\frac{20.57}{R}$

${\left(\frac{1}{{\lambda }_{\min }}\right)}_{\text{Paschen}}=R\left(\frac{1}{3^2}-\frac{1}{{\infty }^2}\right)\Rightarrow {\left({\lambda }_{\min }\right)}_{\text{Paschen}}=\frac{9}{R}$

So, Paschen series wavelengths vary between $\frac{9}{R}$ to $\frac{20.57}{R}$

So, it is clear that wavelength ranges of none of the series overlap with one another. 

Option (b) is wrong. Option (d) is correct.